Solving the Age Word Problem

This problem involves finding the present age of Pinaki and Bhaswati using the given relationships between their ages now and in the future. We can use algebraic equations to represent these relationships and solve for the unknown ages.

Setting Up the Equations for Pinaki and Bhaswati's Ages

Let's define variables for the present ages:

• Let Pinaki's present age be \(P\) years.
• Let Bhaswati's present age be \(B\) years.

From the first statement, "Pinaki is 9 years younger than Bhaswati", we can write the equation:

\(P = B - 9\)

This can also be written as \(B = P + 9\). This equation relates their present ages.

Ages Thirteen Years Hence

Now consider the ages thirteen years from now:

• Pinaki's age thirteen years hence will be \(P + 13\) years.
• Bhaswati's age thirteen years hence will be \(B + 13\) years.

The second statement says, "Thirteen years hence Bhaswati will be 1.2 times as old as Pinaki". We can write this as the equation:

\(B + 13 = 1.2 \times (P + 13)\)

Solving the Equations to Find Pinaki's Present Age

We have two equations:

1. \(B = P + 9\)
2. \(B + 13 = 1.2 \times (P + 13)\)

We can substitute the expression for \(B\) from equation (1) into equation (2):

\((P + 9) + 13 = 1.2 \times (P + 13)\)

Simplify the left side:

\(P + 22 = 1.2 \times (P + 13)\)

Distribute 1.2 on the right side:

\(P + 22 = 1.2P + 1.2 \times 13\)

\(P + 22 = 1.2P + 15.6\)

Now, we need to isolate \(P\). Subtract \(P\) from both sides:

\(22 = 1.2P - P + 15.6\)

\(22 = 0.2P + 15.6\)

Subtract 15.6 from both sides:

\(22 - 15.6 = 0.2P\)

\(6.4 = 0.2P\)

To find \(P\), divide 6.4 by 0.2:

\(P = \frac{6.4}{0.2}\)

\(P = \frac{64}{2}\)

\(P = 32\)

So, Pinaki's present age is 32 years.

Verifying the Solution

Let's check if this answer satisfies the original conditions:

• Pinaki's present age \(P = 32\) years.
• Bhaswati is 9 years older than Pinaki, so Bhaswati's present age \(B = P + 9 = 32 + 9 = 41\) years.

Thirteen years hence:

• Pinaki's age will be \(32 + 13 = 45\) years.
• Bhaswati's age will be \(41 + 13 = 54\) years.

Is Bhaswati's age 1.2 times Pinaki's age thirteen years hence?

\(1.2 \times 45 = 54\)

Yes, the ages \(54\) and \(45\) satisfy the condition \(54 = 1.2 \times 45\). The solution is correct.

Let's summarize the ages in a table:

The calculated present age for Pinaki is 32 years.

Revision Table: Age Word Problems

Additional Information on Solving Word Problems

Solving word problems, especially those involving ages, often follows a systematic approach:

• Read Carefully: Understand the relationships and conditions given in the problem. Identify what needs to be found.
• Assign Variables: Use variables (like \(x\), \(y\), \(P\), \(B\)) to represent the unknown quantities, usually the present ages.
• Formulate Equations: Translate the sentences describing relationships between ages into mathematical equations. Pay attention to phrases like "younger than", "older than", "times as old", "years hence", "years ago".
• Solve the Equations: Use algebraic methods (substitution, elimination) to solve the system of equations you have created.
• Check the Solution: Substitute the values you found back into the original word problem statements to ensure they satisfy all conditions. This step is crucial to avoid errors.
• State the Answer: Clearly write down the answer in the context of the original question.

Age word problems are a common type of question in quantitative aptitude tests and require careful translation of verbal statements into algebraic expressions.